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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Shrinking countable decompositions of $ S\sp{3}$

Authors: Richard Denman and Michael Starbird
Journal: Trans. Amer. Math. Soc. 276 (1983), 743-756
MSC: Primary 57N12; Secondary 54B15
MathSciNet review: 688975
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Abstract: Conditions are given which imply that a countable, cellular use decomposition $ G$ is shrinkable. If the embedding of each element in $ G$ has the bounded nesting property, defined in this paper, then $ {S^3}/G$ is homeomorphic to $ {S^3}$. The bounded nesting property is a condition on the defining sequence of cells for an element of $ G$ which implies that $ G$ satisfies the Disjoint Disk criterion for shrinkability [ $ {\mathbf{S1}}$, Theorem 3.1]. From this result, one deduces that countable, star-like equivalent use decompositions of $ {S^3}$ are shrinkable--a result proved independently by E. Woodruff [ $ {\mathbf{W}}$]. Also, one deduces the shrinkability of countable bird-like equivalent use decompositions (a generalization of the star-like result), and the recently proved theorem that if each element of a countable use decomposition $ G$ of $ {S^3}$ has a mapping cylinder neighborhood, then $ G$ is shrinkable [ $ {\mathbf{E}}$; $ {\mathbf{S1}}$, Theorem 4.1; $ {\mathbf{S}}$-$ {\mathbf{W}}$, Theorem 1].

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Keywords: Countable cellular decomposition, bounded nesting, star-like, bird-like
Article copyright: © Copyright 1983 American Mathematical Society